1. Deformation of Nanotubes Yang Xu and Kenny Higa MatSE 385 http://hp720.ceg.uiuc.edu/~rotkin/nanotube/nems-1.html

2. Introduction We have adopted a simple model of a nanoelectromechanical switch [1] We have simulated the effect of introducing a defect by deleting atoms in a cylindrical region Presentation: Pull-in voltages of carbon nanotube-based nanoelectromechanical switches. Marc Dequesnes, et. al

3. Algorithm overview Hybrid tight-binding method [2] combines features of both ab initio and MD methods

4. Schrodinger equation Wave function Charge density Poisson solver Potential distribution Initial guess potential Solve Schrodinger equation self-consistently Iteration to get self- consistent results Tight-binding approximation: consider interactions between layers (cross-sectional slices) of the nanotube Interaction potential is non-zero only for nearest neighbors Copyright V. H. Crespi. Distributed under the Open Content License (http://opencontent.org/opl.shtml).

5. Schrodinger equation Wave function Charge density Poisson solver Potential distribution Initial guess potential Solve Schrodinger equation self-consistently * Construct the Potential Matrix Iteration Only considering potential terms of Hamiltonian Kinetic part which is relative to temperature (E k =3/2nKT) is constant in our model Image force e  m-n EFEF EcEc Metal Nanotube * Schottky barrier potential is included

6. Schrodinger equation Wave function Charge density Poisson solver Potential distribution Initial guess potential Solve Schrodinger equation self-consistently * Tight-Binding Approximation Iteration We got a block-diagonal matrix eigenvalue problem

7. Solve Schrodinger equation self-consistently Finite-element method used to solve Poisson equation to determine potential field Charge is non-zero only in nanotube giving sparse matrix system Iterate until self-consistent solution is obtained Schrodinger equation Wave function Charge density Poisson solver Potential distribution Initial guess potential Iteration

8. Quantum results

9. Electrostatic Force applied on the nanotube

10. Molecular dynamics Initialize velocities from Maxwell- Boltzmann distribution Velocity Verlet algorithm used to update carbon atom positions Particle motion influenced by van der Waals interactions, covalent bonding, electric field

11. Van der Waals interactions Nanotube interactions with graphite plane important on nanoscale [1] Modeled using Lennard-Jones potential Existing code used to calculate van der Waals force per unit length [1] Horizontal forces neglected

12. Tersoff Potential Tersoff potential has been successfully for carbon bonding in graphite, diamond [3] Realistic model of bond energies and lengths Sum over nearest neighbors Attractive and repulsive forms similar to Morse potential but considers bond order

13. Electric field Electric charge per length determined from quantum calculations Force due to external field calculated from analytical expression Force due to induced electric field calculated using image charges Horizontal forces neglected

14. Some pictures

16. Another picture

17. One last picture

18. Conclusion Quantum effects are important at the nanotube ends Simulating 6e-11 seconds takes around 10 hours We have not generated enough data for quantitative conclusions Nanotube has not made contact with graphite plate Simulations suggest that nanotubes tend to bend most near point of attachment

19. References [1] Desquesnes, M., Rotkin, S. V., and Aluru, N. R. “Calculation of pull-inn voltages for carbon-nanotube-based nanoelectromechanical switches”, Nanotechnology (13) 120-131, 2002. [2] Clementi E. “Ab initio computations in atoms and molecules”, (reprinted from IBM Journal of Research and Development 9, 1965), IBM J. Res. Dev. 44 (1- 2:228-245, 2000. [3] Brenner, D. W. “Emperical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films”, Physical Review B. Volume 42, Number 15: 9458-9471, 1990. Special thanks to Yan Li, Zhi Tang, Rui Qiao, and Marc Dequesnes for their advice and for writing the code that formed the basis for our project.